Press release

May 27, 2022

Keio University

Development of Highly Accurate Integration AI Using Deep Learning－Highest Score Ever Achieved by AI in Symbolic Integration Tests－A research group comprising Hazumi Kubota and Yuta Tokuoka (of the same affiliation attime of study) of the Keio University Graduate School of Science and Technology, andAssistant Professor Takahiro Yamada, and Professor Akira Funahashi of the Faculty ofScience and Technology have developed AIs capable of predicting the integrated function(primitive function) from the input function subject to integration (integrand), afterremarking on the similarities between the mathematical processing of integrals learned inhigh school and those for language translation using AIs, which have seen remarkableprogress in recent years. Taking inspiration from the fact that it is possible to determine theaccuracy of integration from taking a differentiated primitive function produced by AI anddetermining whether it is consistent with the integrand, the group developed a method for
creating and training diverse AIs, and deploying those AIs capable of producing the correct
answer from among these. These results demonstrate that the developed AIs were capable ofintegration to an accuracy of 99.79%, the highest ratio ever achieved when compared topreviously developed integration tools such as Mathematica and other machine learning
methods. Furthermore, investigating the characteristics of the functions learned by the AI
revealed that each of the constructed models had functions to which they were suited andthose to which they were less suited when performing integrations. Accordingly, these models
were capable of facilitating high degrees of accuracy by solving integrations in a crosssubsidized manner. Integration is an essential process for simulations in control engineering,systems biology, and other domains, and these results are anticipated to contribute toincreasingly accurate simulations in these fields.
Prior to their publication in the journal IEEE Access, a preliminary online version of theseresearch results was published on the journal’s website on April 29.
1. Main Points of Research・Researchers implemented an AI based on Long Short-Term Memory (LSTM)*1 and transformer
models*2 to output primitive functions with integrands as input, formulating an algorithm toselect the AI models whereby the functions subject to integration were consistent with theoutput primitive functions when differentiated.
・AI models were trained in the task of integration to discern the primitive function which would
consist of an equation obtained by multiplying or dividing an elementary function by up to five.
・The results demonstrated that, after training, the AIs were capable of performing integrations
to an accuracy of 99.79% when supplied with integrands in which they had not been trained,
and that this accuracy was more significant than that produced when the same integrations
were solved using Mathematica, a previously-developed integration tool, or methods based on

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machine learning.
・ By analyzing the conversion process from AI-processed integrand functions to primitive
functions, researchers demonstrated that each AI had its respective strengths and weaknesses.
For example, if a particular AI was poor at integrating long integrand functions, working in across-subsidized manner with other AIs enabled the developed AIs to achieve high accuracy.
2. Background of Research
Symbolic integration, which is learned in high school, is not limited to the field of mathematics,
but rather represents an essential process for simulations to predict the dynamics of phenomenaof interest in a wide range of fields, including computer science, control engineering, mechanicalengineering, and systems biology. In recent years, Dr. Guillaume Lample et al. constructed an AIbased transformer model, which takes integrands as its input and the primitive function as output,
demonstrating its capability in integrating functions that could not previously be solved using
computers [1]. However, the method developed by Dr. Lample et al. was not capable ofconstructing an AI optimized for integration, as the AI was incapable of processing informationcontained in equations such as the order of operations.
To overcome this, the present research group expressed input and output forms as an Abstract
Syntax Tree (AST)*3, allowing for the mathematical information relevant to integrations to betaken into account. They also constructed a wide range of AIs, including LSTM-based AIs which
they anticipated would rigorously learn the order of operations. In addition, the group focused on
the fact that, for integrations, accuracy can be determined by differentiating the primitive function
outputs using AI, and comparing this with the input integrand, creating an algorithm to selectthe accurate results from among the broad range of AIs created.
3. Content of Research and Results
First, researchers prepared a set of 9,697
functions as learning data. These data consist of
created primitive functions obtained bymultiplying or dividing up to five elementaryfunctions ( 𝑥𝑥 or sin 𝑥𝑥 etc.) and the paired
integrand generated by differentiating createdprimitive functions. They developed AI which
took this prepared integrand as input with theprimitive function as output. Eight AIs weredeveloped and trained in integration problemsprepared for each—each AI with a different
combination of the following three approaches: 1)
Inputting the functions as an Abstract Syntax Tree
Fig. 1 Diagram of AI performing(AST) or a word set of strings 2) Using Polish notation
or reverse Polish notation read protocol for integrands 3) Using AI based on either LSTM or atransformer model. Furthermore, the team constructed an algorithm to select from these eight
AIs those that were capable of producing a mathematical function consistent with the integrandafter the primitive function was differentiated (Fig.1).

2/4Using the trained AI to solve integration problems which had not been assigned during machinelearning, researchers demonstrated that integration to an accuracy of 99.7% of correct responses
could be achieved, an indicator of the consistency of the correct primitive function and the functionof the AI output. In addition, comparisons using the same problems solved by the previously
developed integration tool Mathematica and AI developed by Dr. Lample, et al, showed that thepresent research group’s method is the most accurate.
Subsequently, the team investigated the characteristics of the integrands on which AIs focusedin order to learn the reasons for the high accuracy achieved by the AIs developed in this study. Asa result, they discovered that each AI had its respective strengths and weaknesses in integrationsuch as an AI input with a simple character string which was more likely to yield errors when
integrating long numerical expressions input as integrands. The method developed by thisresearch group, whereby they selected the AI that answered correctly, has led to higher accuracies
being achieved by means of cross-subsidizing the strengths and weaknesses of respective AIs.
4. Future Developments
The results of this research have shown that AI can perform highly-accurate integrations.
Furthermore, detailed analyses of the functions used by the AI for integration reveal that each
AI has its own respective strengths and weaknesses in solving integration problems, dependingon the input method and the order it carried out operations. It can be anticipated that theapplication of the AI developed will lead to more accurate predictions in the future in a wider
range of fields by using them in integration simulations, which in recent years have been used to
predict the spread of COVID-19, among other applications.
Acknowledgments
This research was funded and supported by the Japan Science and Technology Agency (JST)JST Strategic Basic Research Program （JPMJCR2011）.
References[1] G.Lample and F.Charton “Deep learning for symbolic mathematics,” in Int. Conf. Learn.
Represent. (ICLR) 2020, Addis Ababa, Ethiopia, Apr. 26-30 2020, pp. 1–24.
Details of Journal Article
Article title: Symbolic Integration by Integrating Learning Models with Different Strengths andWeaknesses
Authors: Hazumi Kubota1; Yuta Tokuoka1; Takahiro G. Yamada1,2; Akira Funahashi1,2
1 Graduate School of Science and Technology, Keio University ; 2 Department ofBiosciences and Informatics, Keio University
Journal: IEEE Access
DOI: 10.1109/ACCESS.2022.3171329

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Glossary and References*1 Long Short-Term Memory (LSTM)
Recurrent neural network capable of learning long-term dependencies of input time series
data (sentences, numerical expressions, etc.)*2 Transformer
Neural network capable of learning latent significance of inputs by automatically focusing onrelationships between elements (words, operators in formulas, etc.)*3 Abstract Syntax Tree (AST)
Data structure to express time series data with a meaningful order such as mathematicalfunctions*Please direct any requests or inquiries on press coverage to the contact information provided
below in advance.
・Inquiries about research
Department of Biosciences and Informatics, Faculty of Science and Technology,
Keio University
Professor Akira Funahashi
TEL: +81 (0)45-566-1797
FAX: +81 (0)45-566-1789
Email: funa@bio.keio.ac.jp
・Inquiries about press release
Keio University Office of Communications and Public Relations (Contact: Sawano)
TEL: +81 (0)3-5427-1541
FAX: +81 (0)3-5441-7640
Email: m-pr@adst.keio.ac.jp
https://www.keio.ac.jp/en/

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